Of course you can use words however you want - but using them in a way which is not their correct definition is simply inaccurate.
I could go around calling the road a tree, calling my house a cake or calling cars penguins - it doesn't legitimise my use of the word - I can't suddenly claim that my car is 100% carbon-free by calling it a penguin.
So, call whatever you wish by whatever name you like, you will be in your own world of fantasy and make believe (and if you are uncertain - look those words up in the dictionary, they haven't been redefined yet!)
The issue in discussion here is that the bizarre choice by the Cambridge Dictionary pretends to legitimise a fantasy and make-believe world view and clearly that is not appropriate.
Fundamentally at the root of this is:
Fact
Man = biological sex = male at birth and thereafter
Woman = biological sex = female at birth and thereafter
Anyone who has been through surgery to change their appearance, or who feels more female / male etc. still doesn't change that basic biological fact, so they still remain a man or woman as they were born.
If they want a new word to describe how they now feel, then (as suggested above) they can invent a new word - but they don't get to use an existing word and redefine it in the process.